Home of the Cubs
Introduction
This is a lesson wherein students will discover a useful tool for solving triangles from a dynamic geometric construction. This should provide a concrete justification for the algebra beyond the teacher’s “because I said so”. There is a Pre-test to assess students prior knowledge which is probably limited and there are resource pages to help guide students through this investigation.
Subject: Mathematics
Topic: Trigonometry
Grade Level: 11 & 12
Student Lesson name and URL: ctap295.ctaponline.org/~pgerrode/student/home.html
Standards Addressed
Mathematics: The Law of
Cosines
Content Strand: Trigonometry
Substrand: 13.0
Goal for Substrand:
(13) Students know the law
of sines and the law of cosines and
apply those laws to solve
problems involving triangles.
Instructional
Objectives
- After experimenting with a dynamic geometric construction that illustrates the law of cosines, students will discover how the three sides and a specified angle of a triangle are related.
- After gathering data and completing the accompanying resource page, students will be able to derive the algebraic representation of the law of cosines.
- After viewing an animation, students will be able to extend and generalize their conclusions from the first activity to include all the altitudes of a triangle.
- Students will be able to apply their knowledge to solve the two types of triangle problems listed below:
- Given two sides and the included angle, students will be able to find the length of the third side of the triangle.
- Given three sides of a triangle, students will be able find the measure of a specified angle.
Student
Activities
Introductory Activity - Triangle Pre-test
Students will encounter application problems on a pre-test for which their previous mathematical knowledge is insufficient. The Law of Sines cannot be used to solve all of these triangles; this hopefully will create the need for students to develop a new tool for solving triangles, namely the Law of Cosines.
Enabling Activity(ies)
1)
This Geometer's Sketchpad activity enables students to investigate and
discover the Law of Cosines as a generalization of the Pythagorean Theorem
to include both acute and obtuse triangles. This activity allows students
to see geometrically how the squares of the sides of any triangle are related
to each other with the aid of a surprising "fudge factor" which results
from the construction of an altitude of the triangle. The corresponding
resource page will guide students through this investigation and concludes
algebraically with the law of cosines.
2) This JAVA Applet "proves" the Law of Cosines by illustrating
how the altitudes of a triangle, as they partition the squares on each
of the sides, create rectangles with equal areas . This result
allows students to extend their knowledge from the first activity above
to consider all the altitudes of a triangle.
Culminating Activity
Students will solve the two application problems encountered on the Applications Resource page. Their solution(s) will be evaluated using a 4-point rubric (see below).
Post-test
Students can revisit the Pre-test to recognize new knowledge gained
Assessment
Four Point Rubric:
| 4 |
Fully accomplishes the purpose of the task. Student work shows full grasp of the central mathematical idea(s). The recorded work communicates thinking clearly using some combination of written, symbolic, or visual means. |
| 3 |
Substantially accomplishes the purpose of the task. Student work shows essential grasp of the central mathematical idea(s). The recorded work in large part communicates the thinking. |
| 2 |
Partially accomplishes the purpose of the task. Student work shows partial but limited grasp of the central mathematical idea(s). The recorded work may be incomplete, misdirected, or not clearly presented. Revision is necessary. |
| 1 |
Makes little or no progress toward accomplishing the task. Shows little or no grasp of the central mathematical idea(s). The recorded work is barely (if at all) comprehensible. The student must restart the problem. |
Results
Results on this chart are for presentation purposes only and will be updated during the Spring of 2002 when a more complete field test can be performed.
Web
Resources & Supplementary Materials
To utilize all aspects of this lesson, you may have to enable your browser to run JAVA, and also you may need some software applications: Word 97 or higher,and Geometer's Sketchpad from Key Curriculum Press. In addition, if you are using Netscape and are having difficulty linking to any of the sketches, try using Internet Explorer instead.
Introductory Activity
Triangle
Pre-test
(a Word document)
Enabling Activities
Law
of Cosines Investigation
(a Geometer's Sketchpad sketch)
with Resource page
(a Word document)
" Proof " of the Law of Cosines (a JAVA Applet)
Culminating Activity (a Word document)
Post-test (Same as Pre-test) (a Word document)
A Personal Reflection (a Powerpoint presentation)
School Location: 1471 Ferry St.
Anderson, CA 96007
Peter Gerrodette,pgerrodette@anderson.k12.ca.us
Last Revised: 06/30/2000